非局部 KP -型方程的广义双线性导数

Abstract

Abstract:

A generalized bilinear transformation method, as an extension of bilinear transformation method due to Hirota, is applied to study the nonlinear dispersive equations with nonlocal terms. With this method, the KP-type equations including the KP-II, BKP and (3+1)-dimensional generalized KP equations are studied and their nonlocal forms are derived. As the conclusions, the solitary wave solutions and approximate solutions of those KP-type equations are constructed by developing the bilinear transformation method.

Key words: Bilinear transform method, Generalized bilinear derivative, Approximate solution, KP equation, BKP equation

CLC Number:

O175.2

Zhao Qian, Yan Lu. Nonlocal KP-type Equations with Generalized Bilinear Derivative[J].Acta mathematica scientia,Series A, 2023, 43(1): 143-158.

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References 40

[1]	Hirota R. Direct methods in soliton theory// Bullough R K and Caudrey P J. Solitons. Berlon: Springer-Verlag, 1980: 157-176
[2]	Hirota R. Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett, 1971, 27(18): 1192-1194 doi: 10.1103/PhysRevLett.27.1192
[3]	Matsuno Y. Bilinear Transformation Method. New York: Academic Press, 1984
[4]	Satsuma J. $N$-soliton solution of the two-dimensional Korteweg-de Vries equation. J Phys Soc Japan, 1976, 40: 286-290 doi: 10.1143/JPSJ.40.286
[5]	Hirota R. Exact solution of the sine-Gordon equations for multiple collsions of solitons. J Phys Soc Japan, 1972, 33: 1459-1463 doi: 10.1143/JPSJ.33.1459
[6]	Hirota R. Exact $N$-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices. J Math Phys, 1973, 14: 810-814 doi: 10.1063/1.1666400
[7]	Hietarinta J. A search for bilinear equations passing Hirota's three-soliton condition. I. KdV-type bilinear equations. J Math Phys, 1987, 28: 1732-1742 doi: 10.1063/1.527815
[8]	Hietarinta J. A search for bilinear equations passing Hirota's three-soliton condition. II. mKdV-type bilinear equations. J Math Phys, 1987, 28: 2094-2101 doi: 10.1063/1.527421
[9]	Hietarinta J. A search for bilinear equations passing Hirota's three-soliton condition. III. Sine-Gordon-type bilinear equations. J Math Phys, 1987, 28: 2586-2592 doi: 10.1063/1.527750
[10]	Hietarinta J. A search for bilinear equations passing Hirota's three-soliton condition. IV. Complex bilinear equations. J Math Phys, 1988, 29: 628-635 doi: 10.1063/1.528002
[11]	Hu X B. Hirota-type equations, soliton solutions, Bäcklund transformations and conservation laws. J Part Diff Eq, 1990, 3: 87-95
[12]	Hu X B. Generalized Hirota's bilinear equations and their soliton solutions. J Phys A, 1993, 26: 465-471
[13]	Hirota R. Fundamental properties of the binary operators in soliton theory and their generalization. //Takeno S. Dynamical Problems in Soliton Systems. Berlin: Springer-Verlag, 1985: 42-49
[14]	Ma W X. Generalized bilinear differential equations. Stud Nonlinear Sci, 2011, 2: 140-144
[15]	Xia Q, Lou S Y. Localized excitations of a generalised Jimbo-Miwa equation. Commun Theor Phys, 2018, 70: 1-6 doi: 10.1088/0253-6102/70/1/1
[16]	Kadomtsev B B, Petviashvili V I. On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl, 1970, 15: 539-541
[17]	Ablowitz M J, Segur H. On the evolution of packets of water waves. J Fluid Mech, 1979, 92: 691-715 doi: 10.1017/S0022112079000835
[18]	Biondini G, Pelinovsky D E. Kadomtsev-Petviashvili equation. Scholarpedia, 2008, 3: 6539-6547 doi: 10.4249/scholarpedia.6539
[19]	Pelinovsky D E, Stepanyants Y A, Kivshar Y S. Self-focusing of plane dark solitons in nonlinear defocusing media. Phys Rev E, 1995, 51: 5016-5026 pmid: 9963213
[20]	Druyma V S. On the analytical solution of the two-dimensional Korteweg-de Vries equation. Sov Phys JETP Lett, 1974, 19: 753-757
[21]	Ablowitz M J, Villarroel J. On the Kadomtsev Petviashvili equation and associated constraints. Stud Appl Math, 1991, 85: 195-213 doi: 10.1002/sapm1991853195
[22]	Freeman N C, Nimmo J J C. Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique. Phys Lett A, 1983, 95: 1-3 doi: 10.1016/0375-9601(83)90764-8
[23]	Ablowitz M, Segur H. Solitons and the Inverse Scattering Transform. Philadephia: Society for Industrial and Applied Mathematics, 1981
[24]	Kodama Y, Williams L. KP solitons and total positivity for the Grassmannian. Invent Math, 2014, 198: 637-699 doi: 10.1007/s00222-014-0506-3
[25]	Date E, Jimbo M, Kashiwara M, Miwa T. Transformation groups for soliton equations: III. Operator approach to the Kadomtsev-Petviashvili equation. J Phys Soc Japan, 1981, 50: 3806-3812 doi: 10.1143/JPSJ.50.3806
[26]	Date E, Jimbo M, Kashiwara M, Miwa T. Transformation groups for soliton equations: IV. A new hierarchy of soliton equations of KP-type. Physica D, 1982, 4: 343-365 doi: 10.1016/0167-2789(82)90041-0
[27]	Hirota R. The Direct Method in Soliton Theory. New York: Cambridge University Press, 2004
[28]	Date E, Kashiwara M, Miwa T. Transformation groups for soliton equations: II. Vertex operators and $\tau$ functions. Proc Japan Acad Ser A Math Sci, 1981, 57: 387-392
[29]	Hirota R. Soliton solutions to the BKP equations: I. The Pfaffian techniqueon. J Phys Soc Japan, 1989, 58: 2285-2296 doi: 10.1143/JPSJ.58.2285
[30]	Hirota R. Soliton solutions to the BKP equations: II. The Integral Equation. J Phys Soc Japan, 1989, 58: 2705-2712 doi: 10.1143/JPSJ.58.2705
[31]	Gilson C R, Nimmo J J C. Lump solutions of the BKP equation. Phys Lett A, 1990, 147: 472-476 doi: 10.1016/0375-9601(90)90609-R
[32]	Loris I, Willox R. Symmetry reductions of the BKP hierarchy. J Math Phys, 1999, 40: 1420-1431 doi: 10.1063/1.532812
[33]	刘青平, Manas M. BKP方程族的 Pfaffian解. 数学年刊, 2002, 23A(6): 693-698
	Liu Q P, Manas M. On the Pfaffian solutions of the BKP hierarchy. Chin Ann Math, 2002, 23A(6): 693-698
[34]	Ma W X, Fan E G. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl, 2011, 61: 950-959 doi: 10.1016/j.camwa.2010.12.043
[35]	Ma W X, Pekcan A. Uniqueness of the Kadomtsev-Petviashvili and Boussinesq equations. Z Naturforsch A, 2001, 66: 377-382 doi: 10.1515/zna-2011-6-701
[36]	Wazwaz A M. Multiple-soliton solutions for a (3+1)-dimensional generalized KP equation. Commun Nonlin Sci Numer Simul, 2012, 17: 491-495 doi: 10.1016/j.cnsns.2011.05.025
[37]	Ma W X, Xia T C. Pfaffianized systems for a generalized Kadomtsev-Petviashvili equation. Phys Scr, 2013, 87: 055003 doi: 10.1088/0031-8949/87/05/055003
[38]	Wu J P, Geng X G. Novel Wronskian condition and new exact solutions to a (3+1)-dimensional generalized KP equation. Commun Theor Phys, 2013, 60: 556-560 doi: 10.1088/0253-6102/60/5/08
[39]	Wazwaz A M, El-Tantawy S A. A new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation. Nonlinear Dyn, 2016, 84: 1107-1112 doi: 10.1007/s11071-015-2555-6
[40]	You F C, Xia T C, Chen D Y. Decomposition of the generalized KP, cKP and mKP and their exact solutions. Phys Lett A, 2008, 372: 3184-3194 doi: 10.1016/j.physleta.2008.01.036

Nonlocal KP-type Equations with Generalized Bilinear Derivative

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